2.7 Near-extremal near-horizon geometries

An important question about near-horizon geometries is the following: how much dynamics of gravity
coupled to matter fields is left in a near-horizon limit such as (23)? We will explore in the following
Sections 2.8 and 2.9 several aspects of the dynamics in the near-horizon limit. In this section, we will
discuss the existence of near-extremal solutions obtained from a combined near-horizon limit and zero
temperature limit. We will discuss in Section 2.8 the absence of non-perturbative solutions in the
near-horizon geometries, such as black holes. In Section 2.9, we will argue for the absence of local bulk
degrees of freedom, and finally in Section 4.4 we will discuss non-trivial boundary dynamics generated by
large diffeomorphisms.

Let us first study infinitesimal perturbations of the near-horizon geometry (25). As a consequence of the
change of coordinates and the necessary shift of the gauge field (23), the near-horizon energy of an
infinitesimal perturbation is related to the charge associated with the generator of the horizon
as follows,

as derived in Sections 2.2 and 2.3. Assuming no magnetic charges for simplicity, the conserved charge is given by
.10
Using the first law of thermodynamics valid for arbitrary (not necessarily stationary) perturbations, the
left-hand side of (76) can be expressed as

Any geometry that asymptotes to (25) will have finite near-horizon energy . Indeed, an infinite
near-horizon energy would be the sign of infrared divergences in the near-horizon geometry and it would
destabilize the geometry. It then follows from (76) – (77) that any infinitesimal perturbation of the
near-horizon geometry (25) will correspond to an extremal black-hole solution with vanishing
Hawking temperature, at least such that . Common usage refers to black-hole
solutions, where as near-extremal black holes. Nevertheless, it should be emphasized that
after the exact limit is taken the Hawking temperature of such a solution is exactly
zero.

We can obtain a near-extremal near-horizon geometry as follows. Starting from a stationary
non-extremal black hole of mass in Boyer–Lindquist coordinates, we perform the near-horizon scaling
limit (23) together with the scaling of the temperature

While the form of the general non-extremal solution would be required to perform that limit in detail, all
examples so far in the class of theories (1), such as the Kerr–Newman–AdS black hole, lead to the following
metric

The near-extremal near-horizon solution (79) is diffeomorphic to the near-horizon geometry in
Poincaré coordinates (25). Denoting the finite temperature coordinates by a subscript and the
Poincaré coordinates by a subscript , the change of coordinates reads as [207, 247, 4, 53]

where

Therefore, the classical geometries are equivalent. However, since the diffeomorphism is singular at the
boundary , there is a distinction at the quantum level. Since the asymptotic time in
near-extremal geometries (79) is different than in extremal geometries (25), fields will be quantized in a
different manner in the two geometries.

Let us now compute the energy of these geometries. Multiplying Eq. (76) by and using
(77) and (78), we get that the energy variation around the near-extremal geometry is given by

where the extremal entropy can be expressed in terms of the near-horizon quantities as (50). We
denote the variation by to emphasize that the energy is not the exact variation of a quantity unless
is constant or is fixed (which would then lead to zero energy). Therefore, the charge
is a heat term, which does not define a conserved energy. Since our derivation of the
formula (84) was rather indirect, we check that it is correct for the Kerr–Newman–AdS family of
black holes by computing the energy variation directly using the Lagrangian charges defined
in [36, 90, 97].